The Triangle Inequality and Its Powerful Variations
Master the fundamental inequality $||z_1| - |z_2|| \leq |z_1 + z_2| \leq |z_1| + |z_2|$ and its applications in solving complex JEE modulus problems.
Why Triangle Inequality Matters in JEE
The Triangle Inequality is one of the most powerful tools in complex numbers, appearing in over 70% of JEE Advanced complex number problems. Mastering its variations will help you:
- Quickly find maximum and minimum values of complex expressions
- Solve challenging modulus equations efficiently
- Understand geometric interpretations of complex operations
- Save 3-5 minutes per problem during the exam
The Fundamental Triangle Inequality
$$||z_1| - |z_2|| \leq |z_1 + z_2| \leq |z_1| + |z_2|$$
Geometric Interpretation:
In any triangle, the length of one side is always less than the sum of the other two sides and greater than their absolute difference.
Proof Outline:
Upper Bound ($|z_1 + z_2| \leq |z_1| + |z_2|$):
• Use $|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + 2Re(z_1\bar{z_2})$
• Apply $Re(z_1\bar{z_2}) \leq |z_1||z_2|$ (Cauchy-Schwarz)
Lower Bound ($||z_1| - |z_2|| \leq |z_1 + z_2|$):
• Replace $z_2$ with $-z_2$ in the upper bound
• Use $|z_1 - z_2| \leq |z_1| + |z_2|$
Finding Maximum and Minimum of $|z_1 + z_2|$
Maximum: $|z_1 + z_2|_{\text{max}} = |z_1| + |z_2|$
Minimum: $|z_1 + z_2|_{\text{min}} = ||z_1| - |z_2||$
When equality occurs:
Maximum: When $z_1$ and $z_2$ have the same argument ($\arg z_1 = \arg z_2$)
Minimum: When $z_1$ and $z_2$ have opposite arguments ($\arg z_1 = \arg z_2 + \pi$)
Example: JEE Advanced 2019
Problem: If $|z| = 2$ and $|w| = 3$, find the maximum and minimum of $|z + w|$
Solution:
• Maximum: $|z| + |w| = 2 + 3 = 5$
• Minimum: $||z| - |w|| = |2 - 3| = 1$
• Range: $1 \leq |z + w| \leq 5$
Finding Maximum and Minimum of $|z_1 - z_2|$
Maximum: $|z_1 - z_2|_{\text{max}} = |z_1| + |z_2|$
Minimum: $|z_1 - z_2|_{\text{min}} = ||z_1| - |z_2||$
When equality occurs:
Maximum: When $z_1$ and $z_2$ have opposite arguments
Minimum: When $z_1$ and $z_2$ have the same argument
Example: JEE Main 2021
Problem: If $|z - 1| = 2$ and $|z - i| = 3$, find the range of $|z|$
Solution Approach:
• Let $z_1 = z - 1$, $z_2 = z - i$
• Use triangle inequality on $|z_1 - z_2| = |(z-1) - (z-i)| = |i-1| = \sqrt{2}$
• Apply $||z_1| - |z_2|| \leq |z_1 - z_2| \leq |z_1| + |z_2|$
🚀 Problem-Solving Strategies
For Maximum/Minimum Problems:
- Identify given moduli values first
- Check if arguments are specified
- Use geometric interpretation when possible
- Remember equality conditions
For Complex Equations:
- Try triangle inequality bounds first
- Check if equality can be achieved
- Use substitution $z = re^{i\theta}$
- Consider geometric representations
Variations 3-5 Available in Full Version
Includes Generalized Triangle Inequality, Reverse Triangle Inequality, and Applications in Solving Challenging Modulus Problems
📝 Quick Self-Test
Try these JEE-level problems to test your understanding:
1. If $|z| = 3$ and $|w| = 4$, find the maximum value of $|z + 2w|$
2. If $|z - 2| = 1$ and $|z - 3| = 2$, find the range of $|z|$
3. Prove that $|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2(|z_1|^2 + |z_2|^2)$
Ready to Master All Triangle Inequality Variations?
Get complete access to all variations with step-by-step video solutions and practice problems